Given $ m \angle CBD = 2x - 5$, $ m \angle ABC = 3x - 15$, and $ m \angle ABD = 30$, find $m\angle CBD$. $B$ $A$ $D$ $C$
Solution: From the diagram, we see that together ${\angle ABC}$ and ${\angle CBD}$ form ${\angle ABD}$ , so $ {m\angle ABC} + {m\angle CBD} = {m\angle ABD}$ Substitute in the expressions that were given for each measure: $ {3x - 15} + {2x - 5} = {30}$ Combine like terms: $ 5x - 20 = 30$ Add $20$ to both sides: $ 5x = 50$ Divide both sides by $5$ to find $x$ $ x = 10$ Substitute $10$ for $x$ in the expression that was given for $m\angle CBD$ $ m\angle CBD = 2({10}) - 5$ Simplify: $ {m\angle CBD = 20 - 5}$ So ${m\angle CBD = 15}$.